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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 0162 4986 3
Grebert, B; Kappeler, T.
Estimates of eigenvalues for Zakharov Shabat system
Asymptotic Anal. 25, No. 3-4, 201-237 (2001)
Primary Classification:
34L20Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions
Secondary Classification:
37K15Integration of completely integrable systems by inverse spectral and scattering methods
35Q55NLS-like nonlinear Schroedinger equations
complex version of the Zakharov-Shabat system; periodic and Dirichlet boundary conditions; asymptotic distribution of eigenvalues;

Nonlinear Schr\"{o}dinger equation may be assigned (or represented
and analyzed in terms of) a Lax pair of operators of the Zakharov
Shabat (in essence: perturbed Dirac operator) type. This context
and related specific range of applications limit, often, the
assumptions and analysis, typically, to the self-adjoint case. The
paper deals with the generalized non-selfadjoint two-by-two
complex Zakharov Shabat linear differential operator, with (in the
Dirac-equation language) ``potential" term assumed to lie within a
certain (suitably weighted) Sobolev space.
The spectrum is still known to be discrete (for both the periodic
and Dirichlet boundary conditions). The suitable asymptotic
expansions of the eigenvalues are available in the self-adjoint
case but not in its present generalization. The authors employ a
nonstandard method (called a Lyapunov-Schmidt type decomposition
in the paper) and offer the two basic theorems on the asymptotic
distribution of the (complex) eigenvalues.
The text is inspired by the recent similar analysis of the linear
Schroedinger operators. This puts this paper in a perspective of a
natural methodical development since, in the later context, the
method is well known under many other names (e.g., as a Loewdin's
projection operator method in perturbation theory and in its
applications in quantum chemistry or as the feshbach's effective
operator method in quantum mechanics and nuclear physics, etc).
Its present application is innovative and leads to a surprisingly
compact picture of the generalized case, with the key idea lying
in a successful elimination of all the ``irrelevant" Fourier
components of the eigenfunctions so that we are left with the
two-by-two ``effective" algebraic eigenvalue problem.
Remarks to the editors:

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